Free · Mathematical relationships

Triangle from Two Sides and Included Angle

Two positive sides and an included angle strictly between 0° and 180° determine one triangle. The cosine rule determines its third side.

Formula & worked example
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Enter the known values in the units shown. Results update as you type.
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Formula and method

With sideA = 3, sideB = 4, angle_c = 90, sideC = 5, area = 6, peri = 12.

c²=a²+b²−2ab cos(C); A=ab sin(C)/2

Worked example

Enter these known values and leave the other values blank.

Side a: 3 cm
Side b: 4 cm
Angle γ: 90 deg

Side c: 5 cm
Angle α: 36.87 deg
Angle β: 53.13 deg
Area: 6 cm²
Perimeter: 12 cm

Assumptions and limitations

Two positive sides and an included angle strictly between 0° and 180° determine one triangle. The cosine rule determines its third side.
Enter enough independent values and clear the quantity being solved. Locks preserve known values; contradictory inputs can make the system inconsistent. Unit controls convert physical quantities rather than merely changing their labels.
Displayed precision is numerical rounding, not an estimate of measurement uncertainty.

Common questions

How do I solve for a missing quantity?

Clear the missing value, enter the independent known values and retain compatible units. A nonunique inverse may require another known value.

What relationship is used?